S3.1 Einstein's Second Revolution

Our goals for learning:

What are the major ideas of general relativity?

What is the fundamental assumption of general relativity?

Spacetime

Special relativity showed that space and time are not absolute.

Instead, they are inextricably linked in a four-dimensional combination called spacetime.

Curved Space

Travelers going in opposite directions in straight lines will eventually meet.

Because they meet, the travelers know Earth's surface cannot be flat—it must be curved.

Curved Spacetime

Gravity can cause two space probes moving around Earth to meet.

General relativity says this happens because spacetime is curved.

Rubber Sheet Analogy

Matter distorts spacetime in a manner analogous to how heavy weights distort a rubber sheet.

Key Ideas of General Relativity

Gravity arises from distortions of spacetime.

Time runs slowly in gravitational fields.

Black holes can exist in spacetime.

The universe may have no boundaries and no center but may still have finite volume.

Rapid changes in the motion of large masses can cause gravitational waves.

Relativity and Acceleration

Our thought experiments about special relativity involved spaceships moving at constant velocity.

Is all motion still relative when acceleration and gravity enter the picture?

Acceleration and Relative Motion

How can your motion be relative if you're feeling a force causing acceleration?

The Equivalence Principle

Einstein preserved the idea that all motion is relative by pointing out that the effects of acceleration are exactly equivalent to those of gravity.

Gravity and Relative Motion

Someone who feels a force may be hovering in a gravitational field. Someone who feels weightless may be in free-fall.

S3.2 Understanding Spacetime

Our goals for learning: What

is spacetime?

What

is curved spacetime?

Dimensions of Space

An object's number of dimensions is the number of independent directions in which movement is possible within the object.

Dimensions of Spacetime

We can move through three dimensions in space (x, y, z).

Our motion through time is in one direction (t).

Spacetime, the combination of space and time, has four dimensions (x, y, x, t).

Perspectives in Space

A book has a definite three-dimensional shape.

But the book looks different in two-dimensional pictures of the book taken from different perspectives.

Similarly, space and time look different from different perspectives in spacetime.

Perspectives in Spacetime

Observers in relative motion do not share the same definitions of x, y, z, and t, taken individually: Space

is different for different observers.

Time

is different for different observers.

Spacetime

is the same for everyone.

Spacetime Diagram of a Car

A spacetime diagram plots an object's position in space at different moments in time.

Worldlines

A worldline shows an object's path through spacetime in a spacetime diagram.

Vertical worldline: no motion

Diagonal worldline: constant-velocity motion

Curved wordline: accelerating motion

Worldlines for Light

Worldlines for light go at 45° angles in diagrams with light-seconds on one axis and seconds on the other.

Worldlines and Relativity

Worldlines look different in different reference frames.

Worldlines and Relativity

But everyone will agree on the "distance" between two different events in spacetime: x2 + y2 + z2 – (ct)2.

Rules of Geometry in Flat Space

A straight line is shortest distance between two points.

Parallel lines stay the same distance apart.

Angles of a triangle add up to 180°.

Circumference of a circle is 2r.

Geometry on a Curved Surface

The straightest lines on a sphere are great circles sharing the same center as the sphere.

Great circles intersect, unlike parallel lines in flat space.

Geometry on a Curved Surface

Straight lines are the shortest paths between two points in flat space.

Great circles are the shortest paths between two points on a sphere.

Rules of Spherical Geometry

A great circle is the shortest distance between two points.

Parallel lines eventually converge.

Angles of a triangle add up to > 180°.

Circumference of circle is < 2r.

Rules of Saddle-Shaped Geometry

A piece of a hyperbola is the shortest distance between two points.

Parallel lines diverge.

Angles of a triangle add up to < 180°.

Circumference of circle is > 2r.

Geometry of the Universe

The universe may be flat, spherical, or saddle-shaped depending on how much matter (and energy) it contains. Flat

and saddle-shaped universes are infinite in extent.

Spherical No

universe is finite in extent.

center and no edge to the universe are necessary in any of these cases.

"Straight" Lines in Spacetime

According to equivalence principle: If

you are floating freely, then your worldline is following the straightest possible path through spacetime.

If

you feel weight, then you are not on the straightest possible path.

What have we learned?

What is spacetime? Spacetime

is the four-dimensional combination of space and time that forms the "fabric" of our universe.

What is curved spacetime? Spacetime

can be curved, just as a piece of paper can be curved.

The

three possible geometries for spacetime are flat, spherical, and saddle-shaped.

The

rules of geometry differ for each of these cases.

S3.3 A New View of Gravity

Our goals for learning: What

is gravity?

What

is a black hole?

How

does gravity affect time?

Gravity, Newton, and Einstein

Newton viewed gravity as a mysterious "action at a distance."

Einstein removed the mystery by showing that what we perceive as gravity arises from curvature of spacetime.

Rubber Sheet Analogy

On a flat rubber sheet: Free-falling objects move in straight lines. Circles

all have circumference 2r.

Rubber Sheet Analogy

Mass of Sun curves spacetime: Free-falling Circles

objects near Sun follow curved paths.

near Sun have circumference < 2r.

Limitations of the Rubber Sheet Analogy

Rubber sheet shows only two dimensions of space.

Path of an orbiting object actually spirals through spacetime as it moves forward in time.

Curvature Near Sun

Sun's mass curves spacetime near its surface.

Curvature Near Sun

If we could shrink the Sun without changing its mass, curvature of spacetime would become greater near its surface, as would strength of gravity.

Curvature Near Black Hole

Continued shrinkage of Sun would eventually make curvature so great that it would be like a bottomless pit in spacetime: a black hole.

Curvature Near Black Hole

Spacetime is so curved near a black hole that nothing can escape.

The "point of no return" is called the event horizon.

Event horizon is a threedimensional surface.

Time in an Accelerating Spaceship

Light pulses travel more quickly from front to back of an accelerating spaceship than in other direction.

Everyone on the ship agrees that time runs faster in front than in back.

Time in an Gravitational Field The effects of gravity are exactly equivalent to those of acceleration. Time must run more quickly at higher altitudes in a gravitational field than at lower altitudes.

What have we learned?

What is gravity?

Gravity arises from curvature of spacetime.

What is a black hole?

Spacetime becomes highly curved around a large mass compressed into a tiny space.

Around a black hole, spacetime becomes so curved that nothing can escape.

How does gravity affect time?

Time runs more slowly at lower altitudes in a gravitational field.

S3.4 Testing General Relativity

Our goals for learning: How

do we test the predictions of the general theory of relativity?

What

are gravitational waves?

Precession of Mercury

The major axis of Mercury's elliptical orbit precesses with time at a rate that disagrees with Newton's laws. General relativity precisely accounts for Mercury's precession.

Gravitational Lensing

Curved spacetime alters the paths of light rays, shifting the apparent positions of objects in an effect called gravitational lensing.

Observed shifts precisely agree with general relativity.

Gravitational Lensing

Gravitational lensing can distort the images of objects.

Lensing can even make one object appear to be at two or more points in the sky.

Gravitational Lensing

Gravity of a foreground galaxy (center) bends light from an object almost directly behind it.

Four images of that object appear in the sky (Einstein's Cross).

Gravitational Lensing

Gravity of foreground galaxy (center) bends light from an object directly behind it.

A ring of light from the background object appears in the sky (Einstein Ring).

Gravitational Time Dilation

Passage of time has been precisely measured at different altitudes.

Time indeed passes more slowly at lower altitudes in precise agreement with general relativity.

Gravitational Waves

General relativity predicts that movements of a massive object can produce gravitational waves just as movements of a charged particle produce light waves.

Gravitational waves have not yet been directly detected.

Indirect Detection of Waves

Observed changes in orbit of a binary system consisting of two neutron stars agree precisely with predictions of general relativity.

Orbital energy is being carried away by gravitational waves.

What have we learned?

How do we test the predictions of the general theory of relativity? Precession

of Mercury

Gravitational

lensing

Gravitational

time dilation

What are gravitational waves? Movements

of massive objects produce wavelike disturbances in spacetime called gravitational waves.

S3.5 Hyperspace, Wormholes, and Warp Drive

Our goals for learning: Where

does science end and science fiction begin?

Shortcuts through Space

If we could somehow build a tunnel through the center of Earth, the trip from Indonesia to Brazil would be much shorter.

Could there be analogous tunnels through spacetime?

Shortcuts through Spacetime

Some mathematical solutions of the equations of general relativity allow for shortcuts called wormholes that are tunnels through hyperspace.

Are Wormholes Really Possible?

Wormholes are not explicitly prohibited by known laws of physics, but there is no known way to make one.

If wormholes exist, then they can be used for time travel.

Time travel leads to paradoxes that some scientists believe should rule out the possibility of wormholes.

What have we learned?

Where does science end and science fiction begin? No

known laws of physics prohibit the shortcuts through spacetime known as wormholes.

However,

wormholes would enable time travel, leading to paradoxes that some believe rule out the possibility of their existence.